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Sep 21, 2013
09/13

by
Mark Mineev-Weinstein

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Using our exact time-depending solutions, we solve the Saffman-Taylor finger selection problem in the absence of surface tension by showing that an arbitrary interface in a Hele-Shaw cell evolves to a single uniformly advancing finger occupying one half of the channel width. This result contradicts the generally accepted belief that surface tension is indispensable for the selection of the one-half-width finger.

Source: http://arxiv.org/abs/patt-sol/9705004v3

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Jun 29, 2018
06/18

by
Oleg Alekseev; Mark Mineev-Weinstein

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A point source on a plane constantly emits particles which rapidly diffuse and then stick to a growing cluster. The growth probability of a cluster is presented as a sum over all possible scenarios leading to the same final shape. The classical point for the action, defined as a minus logarithm of the growth probability, describes the most probable scenario and reproduces the Laplacian growth equation, which embraces numerous fundamental free boundary dynamics in non-equilibrium physics. For...

Topics: Exactly Solvable and Integrable Systems, High Energy Physics - Theory, Nonlinear Sciences,...

Source: http://arxiv.org/abs/1604.06822

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Jul 20, 2013
07/13

by
Mark Mineev-Weinstein; Oleg Kupervasser

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We study the exact non-singular zero-surface tension solutions of the Saffman-Taylor problem for all times. We show that all moving logarithmic singularities a_k(t) in the complex plane \omega = e^{i\phi}, where \phi is the stream function, are repelled from the origin, attracted to the unit circle and eventually coalesce. This pole evolution describes essentially all the dynamical features of viscous fingering in the Hele-Shaw cell observed by Saffman and Taylor [Proc. R. Soc. A 245, 312...

Source: http://arxiv.org/abs/patt-sol/9902007v3

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Sep 19, 2013
09/13

by
Mark Mineev-Weinstein; Anton Zabrodin

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The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain. The Laplacian growth equation itself is the quasiclassical version of the string equation that selects the solution to the hierarchy.

Source: http://arxiv.org/abs/solv-int/9912012v1

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Sep 22, 2013
09/13

by
Uriel Frisch; Mark Mineev-Weinstein

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It is shown that the generalizations to more than one space dimension of the pole decomposition for the Burgers equation with finite viscosity and no force are of the form u = -2 viscosity grad log P, where the P's are explicitly known algebraic (or trigonometric) polynomials in the space variables with polynomial (or exponential) dependence on time. Such solutions have polar singularities on complex algebraic varieties.

Source: http://arxiv.org/abs/nlin/0205049v3

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Sep 21, 2013
09/13

by
Giovani L. Vasconcelos; Mark Mineev-Weinstein

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A new general class of exact solutions is presented for the time evolution of a bubble of arbitrary initial shape in a Hele-Shaw cell when surface tension effects are neglected. These solutions are obtained by conformal mapping the viscous flow domain to an annulus in an auxiliary complex-plane. It is then demonstrated that the only stable fixed point (attractor) of the non-singular bubble dynamics corresponds precisely to the selected pattern. This thus shows that, contrary to the established...

Source: http://arxiv.org/abs/1301.0058v4

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Sep 22, 2013
09/13

by
Mark Mineev-Weinstein; Mihai Putinar; Razvan Teodorescu

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Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this...

Source: http://arxiv.org/abs/0805.0049v2

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Sep 21, 2013
09/13

by
Dmitry Khavinson; Mark Mineev-Weinstein; Mihai Putinar

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The planar elliptic extension of the Laplacian growth is, after a proper parametrization, given in a form of a solution to the equation for area-preserving diffeomorphisms. The infinite set of conservation laws associated with such elliptic growth is interpreted in terms of potential theory, and the relations between two major forms of the elliptic growth are analyzed. The constants of integration for closed form solutions are identified as the singularities of the Schwarz function, which are...

Source: http://arxiv.org/abs/0901.3126v1

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Sep 22, 2013
09/13

by
Mark Mineev-Weinstein; Paul B. Wiegmann; Anton Zabrodin

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We establish the equivalence of a 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity.

Source: http://arxiv.org/abs/nlin/0001007v2

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Sep 19, 2013
09/13

by
Mark Mineev-Weinstein; Gary D. Doolen; John E. Pearson; Silvina Ponce Dawson

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Within a class of exact time-dependent non-singular N-logarithmic solutions (Mineev-Weinstein and Dawson, Phys. Rev. E 50, R24 (1994); Dawson and Mineev-Weinstein, Phys. Rev. E 57, 3063 (1998)), we have found solutions which describe the development and pinching off of viscous droplets in the Hele-Shaw cell in the absence of surface tension.

Source: http://arxiv.org/abs/patt-sol/9912006v1